C H A P T E R
I
I N T R O D U C T I O N
1.1
LIQUID CRYSTALS
The
term liquid crystal describes a state of matter that
exists between the anisotropic crystal and the isotropic liquid.
The
liquid
anisotropy
crystalline or mesomorphic substances exhibit strong
in certain properties, yet maintain a certain extent
of fluidity. Liquid crystals were discovered by Reinitzer (1888),
but
the
first
detailed observations
characterlsing these phases
were made by Lehmann (1890).
A
highly anisotropic shape
of the molecules is necessary
for mesomorphism, like for example the long and narrow rod-like
or disc-like shape. The transition to the mesophase can take place
either due to purely thermal processes (thermotropic mesomorphism)
or due to the influence of solvents (lyotropic mesomorphism). This
thesis deals with the study of only thermotropic mesophases.
1.2
THERMOTROPIC LIQUID CRYSTALS
The thermotropic liquid crystals composed of rod-like mole-
cules have been known for a long time. However liquid crystals
composed
of
(Chandrasekhar
disc-shaped molecules were
et al.
only recently discovered
1977). Liquid crystals made up of rod-like
molecules can be classified broadly into three types of phases,
viz., nematic, cholesteric and smectic (Fig. 1.1).
C H A P T E R
I
I N T R O D U C T I O N
1.1
LIQUID CRYSTALS
The
term liquid crystal describes a state of matter that
exists between the anisotropic crystal and the isotropic liquid.
The
liquid
anisotropy
crystalline or mesomorphic substances exhibit strong
in certain properties, yet maintain a certain extent
of fluidity. Liquid crystals were discovered by Reinitzer (1888),
but
the
first
detailed observations
characterising these phases
were made by Lehmann (1890).
A
highly anisotropic shape
of the molecules is necessary
for mesomorphism, like for example the long and narrow rod-like
or disc-like shape. The transition to the mesophase can take place
either due to purely thermal processes (thermotropic mesomorphlsm)
or due to the influence of solvents (lyotropic mesomorphism). This
thesis deals with the study of only thermotropic mesophases.
1.2
THERMOTROPIC LIQUID CRYSTALS
The thermotropic liquid crystals composed of rod-like mole-
cules have been known for a long time. However liquid crystals
composed
of disc-shaped
(Chandrasekhar et al.
molecules were only recently discovered
1977). Liquid crystals made up of rod-like
molecules can be classified broadly into three types of phases,
viz., nematic, cholesteric and smectic (Fig. 1.1).
C H A P T E R
I
I N T R O D U C T I O N
1.1
LIQUID CRYSTALS
The
term liquid crystal describes a state of matter that
exists between the anisotropic crystal and the isotropic liquid.
The
liquid
crystalline or mesomorphic substances exhibit strong
anisotropy in certain properties, yet maintain a certain extent
of fluidity. Liquid crystals were discovered by Reinitzer (1888),
but
the
first
detailed observations characterising these
phases
were made by Lehmann (1890).
A
highly anisotropic shape of the
molecules is necessary
for mesomorphism, like for example the long and narrow rod-like
or disc-like shape. The transition to the mesophase can take place
either due to purely thermal processes (thermotropic mesomorphlsm)
or due to the influence of solvents (lyotropic mesomorphism). This
thesis deals with the study of only thermotropic mesophases.
1.2
THERMOTROPIC LIQUID CRYSTALS
The thermotropic liquid crystals composed of rod-like mole-
cules have been known for a long time. However liquid crystals
composed
of
disc-shaped molecules were only recently discovered
(Chandrasekhar et al.
1977). Liquid crystals made up of rod-like
molecules can be classified broadly into three types of phases,
viz., nematic, cholesteric and smectic (Fig. 1.1).
The
nematic
phase
(Fig.
l.la)
is
characterised
by
a
long
r a n g e o r i e n t a t i o n a l o r d e r of t h e molecules b u t w i t h o u t any t r a n s l a t i o n a l order.
mately
The long a x e s o f
parallel
defines
t h e molecules a r e a l i g n e d a p p r c x i -
t h e average
direction
%, c a l l e d t h e d i r e c t o r ,
A vector
t o one a n o t h e r .
of
orientation
of
these long axes.
and
?i
'-% a r e e q u i v a l e n t .
The d i r e c t o r i s found t o be a p o l a r , i . e . , +
The mesophase i s o p t i c a l l y p o s i t i v e and u n i a x i a l .
The c h o l e s t e r i c mesophase i s b a s i c a l l y a . n e m a t i c t y p e ,
with
a screw a x i s superimposed normal t o t h e p r e f e r r e d molecular d i r e c tion
(Fig.
I t c o n s i s t s of o r i e n t a t i o n a l l y o r d e r e d o p t i c a l l y
I l b
a c t i v e a n i s o t r o p i c molecules,
the
o r t h o g o n a l t o t h e d i r e c t o r . The
t h e d i r e c t o r has turned
'p'.
tive
director
distance
t w i s t i n g about an a x i s
between
in
layers
which
through an a n g l e of 2n d e f i n e s t h e p i t c h
Over a c e r t a i n s p e c t r a l range c h o l e s t e r i c phases show a s e l e c reflection
of
circularly
polarized
light.
The
cholesteric
phase i s o p t i c a l l y n e g a t i v e .
The
smectic
phase
has a
layered
structure.
Within e a c h o f
t h e s e l a y e r s v a r i o u s molecular arrangements a r e p o s s i b l e .
The simp-
l e s t t y p e s a r e s m e c t i c A and C.
I n t h e s m e c t i c A phase ( F i g .
the
parallel
molecules
a r e approximately
t o one
another
and
1 . 1 ~ )
are
a r r a n g e d i n l a y e r s , w i t h t h e long- axes p e r p e n d i c u l a r t o t h e l a y e r .
Within t h e l a y e r t h e c e n t r e s of g r a v i t y of t h e molecules a r e s i t y a t e d a t random and
t h e molecules have a
relatively
m o b i l i t y . The s t r u c t u r e of t h e s m e c t i c C phase ( F i g .
t e d t o t h a t of t h e smectic A phase, i . e . ,
high d e g r e e o f
l . l d ) is rela-
i t c o n s i s t s of e s s e n t i a l l y
Figure 1 . 1
Schematic representation o f the molecular arrangement
in different liquid crystalline phases. (a) Nematic,
( b ) Cholesteric, (c) Smectic A , and ( d ) Smectic C.
p a r a l l e l molecules i n t h e l a y e r s , b u t t h e s e a r e t i l t e d i n a p a r t i c u l a r d i r e c t i o n with r e s p e c t t o t h e l a y e r normal. There a r e v a r i o u s
other
types of
smectics,
smectics B,
E,
D,
F,
G,
H and I which
molecules
and t h e phases
a r e more o r d e r e d t h a n A and C.
Examples o f
two t y p i c a l
'rod- like'
e x h i b i t e d by them a r e g i v e n i n F i g . 1.2.
I n some systems
t h e s m e c t i c phase.
t h e nematic phase r e a p p e a r s on c o o l i n g from
T h i s lower t e m p e r a t u r e nematic phase i s c a l l e d
t h e r e e n t r a n t nematic phase.
D i s c o t i c Liquid C r y s t a l s
As
mentioned
before,
thermotropic
is
mesomorphism
found
to
occur i n compounds made up o f disc- shaped molecules a l s o . I n t h e s e
systems t h e d i r e c t o r i s p a r a l l e l ' t o t h e s h o r t a x e s o f t h e m o l e c u l e s .
The
different
kinds
of
mesomorphic
phases
formed
with
disc- like
compounds a r e :
1
the
columnar
phases
(D)
(Chandrasekhar
et
al.,
1977;
Tinh
e t a l . , 1978; L e v e l u t e t a l . 1979)
2
Lhe nematic phase ( N D ) (Tinh e t a l . 1979), and
3
t h e t w i s t e d nematic phase (N"D
The molecular arrangement
( D e s t r a d e e t a l . , 1980).
i n these
phases i s
shown i n
Fig.
1 . 3 . ' T h e columnar phases a r e formed of columns o f s t a c k e d d i s c - l i k e
m o l e c u l e s . Four d i f f e r e n t t y p e s o f columnar phases have been i d e n t i fied
(Figs.
1.3a-1.3d).
These
have
been
classified
taking
into
Crystal
23'~
Nematic 4 8 ' ~ Isotropic
p-methoxy benzylidene - p!- butylaniline
(MBBA)
(a)
Crystal
113'~ Smectic G 144.5'~ Smectic C 172'~
SmecticA 199'~ Nematic
236.5'~ Isotropic
Terephthal - bis ( -p-butylaniline)( TBBA)
4
Figure 1.2
Typical examples o f compounds made up o f rod-like molecules
which exhibit liquid crystalline phases
Figure 1.3
Schematic representation of the molecular arrangement in discotic
liquid crystals. (a) Columnar phase,
(b)
hexagonal,
and
(c) rectangular modifications of the columnar structure. (d) tilted
columnar phase, (e) Nd phase, (f) twisted nematic (N;) phase.
account
and
the regular
considering
the
i.e.,
columns,
( 0 ) and
irregular
symmetry
hexagonal
(h)
(d) stacking of
of
the
two
dimensional
or
rectangular
t y p e s o f columnar phases t h u s o b t a i n e d a r e D
A t i l t e d p h a s e where t h e d i s c s a p p e a r
(r).
ho'
t o be
the discs
lattice
The
different
a n d Dro
Dhd' D r d
tilted
t o t h e column a x i s i s a l s o p r e s e n t a n d i s c a l l e d t h e D
with
t
of
.
respect
phase (Fig.
1.3d).
The
phase
ND
is an o r i e n t a t i o n a l l y ordered
t h e d i s c s w i t h no long range
contrast
to
the
classical
translational order
nematic
of
arrangement
of
1.3e).
In
(Fig.
rod- like molecules,
the
ND
p h a s e i s o p t i c a l l y u n i a x i a l and n e g a t i v e . The t w i s t e d n e m a t i c p h a s e
N*D is s i m i l a r t o t h e c h o l e s t e r i c p h a s e ( F i g . 1 . 3 f ) . The ND p h a s e
i s q u i t e f l u i d w h i l e t h e columnar p h a s e s a r e v e r y v i s c o u s .
The r e e n t r a n t phenomenon h a s a l s o been o b s e r v e d i n d i s c - l i k e
mesogens
(Tinh
reentrant N
D
e t al.
1984).
Both
reentrant
columnar
phases and
p h a s e s have been o b t a i n e d .
Examples o f
two t y p i c a l d i s c o t i c compounds i s g i v e n i n F i g .
1.4.
Most
phase.
of
Hence
the
the
results
in
properties
this
of
thesis pertain
this
phase w i l l
t o the nematic
be
described
in
d e t a i l i n t h e following section.
1.3
PROPERTIES OF THE NEMATIC PHASE
The
nematic
p h a s e is more o r d e r e d
than t h e i s o t r o p i c phase
Crystal 80.2' C Mesophase 8 6 . 2 ' ~ Isotropic
83.6* C Mesophose
Benzene - hexa - n - heptanoate
.
Crystal 168' C Nematic D 251 5
OC
Isotropic
Hexa Heptyloxy Benzoote of Triphenylene
Figure 1 . 4
Typical examples of compounds made up o f disc-like molecules which exhibit liquid crystalline phases.
and has a lower symmetry. We can therefore define an order parameter, viz., the orientational order parameter which exists in
the nematic phase but vanishes in the isotropic phase.
In
principle a
complete description of the nematic
requires a hierarchy of
phase
n-particle distributions to be specified.
But for simplicity only a singlet distribution is taken into account
in the mean field approximation. The state of alignment of the
molecules in the medium is then specified by an orientational distribution function. If the molecules are considered to be cylindrically
symmetric, the medium is uniaxial and if 8
is the angle made by
the long axis of the molecule with the director, the distribution
function can be written as f(cos 8 ) .
It is difficult to determine even the singlet distribution
function completely, but the first few coefficients of an expansion
in Legendre polynomials of the distribution function are accessible
by
experiment.
Only the even coefficients are non-zero because
of the apolar character of the director.
f(cos 8 ) can be expanded
as
where the coefficients < P
28
> a r e the order parameters.
< P > can
2
be measured by various techniques like NMR, optical birefringence,
Infrared dichroism measurements, etc.
This is often referred to
as 'the order parameter' and is denoted by S.
< P > is more complicated. Depolarized
4
Raman scattering technique seems to be the most convenient (Jen
The measurement of
et al. 1977). Not many measurements of <Po> are available.
If all the molecules are aligned along
such that 8
= 0,
2
then tcos 8 > = 1 and S = 1. If the molecules are randomly distributed in all directions as in the isotropic phase, all values of
are equally probable and <cos2 O> = 1/3 giving S = 0. The nematicisotropic ( N - I ) transition is of first order.
1.4
MOLECULAR THEORIES
If a fluid of hard rods is considered with no forces between
them, other than the one preventing their interpenetration, at
low densities the rods can orient randomly and the fluid is isotropic. As the density is increased it is difficult for the rods to
orient in all possible directions and the fluid undergoes a transition to the anisotropic phase. This transition was theoretically
treated by Onsager (1949). The excess free energy relative to an
ideal gas has to be evaluated using the orientational distribution
function of the rods. Onsager made a virial expansion of the free
energy and retained only the second virial coefficient. This approximation is valid only for very long rods with a length to breadth
ratio
= 100. Zwanzig ( 1 9 6 3 ) evaluated the higher virial coefficients
by restricting the molecules to take up only 3 mutually perpendi-
a s ' t h e o r d e r p a r a m e t e r ' a n d i s d e n o t e d by S.
The
Raman
measurement
scattering
< P4 > i s more c o m p l i c a t e d .
of
technique
seems
to
be
the
most
Depolarized
convenient
(Jen
e t a l . 1 9 7 7 ) . Not many measurements o f < P > a r e a v a i l a b l e .
4
If
all
2
then <cos 8 >
buted
8
such t h a t 8
t h e molecules a r e aligned along
:
1 and S = 1.
= 0,
I f t h e m o l e c u l e s are randomly d i s t r i -
i n a l l d i r e c t i o n s a s i n t h e i s o t r o p i c p h a s e , a l l v a l u e s of
2
a r e e q u a l l y p r o b a b l e and < c o s 82 = 1 / 3 g i v i n g S
= 0 . The n e m a t i c -
i s o t r o p i c ( N - I ) t r a n s i t i o n is o f f i r s t order.
1.4
MOLECULAR THEORIES
If a f l u i d o f h a r d r o d s i s c o n s i d e r e d w i t h n o f o r c e s between
them,
other
than
the
one
preventing
their
interpenetration,
at
low d e n s i t i e s t h e r o d s c a n o r i e n t randomly a n d t h e f l u i d i s i s o t r o pic.
A s the density is increased it is d i f f i c u l t f o r the rods t o
o r i e n t i n a l l p o s s i b l e d i r e c t i o n s and t h e f l u i d u n d e r g o e s a t r a n s i tion
to
treated
the anisotropic
by O n s a g e r
phase.
(1949).
This
transition
was
theoretically
The e x c e s s f r e e e n e r g y r e l a t i v e t o a n
i d e a l gas h a s t o be e v a l u a t e d u s i n g t h e o r i e n t a t i o n a l d i s t r i b u t i o n
function
of t h e rods.
Onsager made a v i r i a l e x p a n s i o n o f t h e f r e e
energy and r e t a i n e d o n l y t h e second v i r i a l c o e f f i c i e n t . T h i s approximation is v a l i d o n l y f o r v e r y long r o d s w i t h a l e n g t h t o b r e a d t h
ratio
= 100.
Zwanzig ( 1 9 6 3 ) e v a l u a t e d t h e h i g h e r v i r i a l c o e f f i c i e n t s
by r e s t r i c t i n g
t h e molecules t o t a k e up o n l y 3 m u t u a l l y perpendi-
cular
orientations.
Flory
(1956)
and
F l o r y a n d Ronca
(1979)
used
a l a t t i c e model and made c a l c u l a t i o n s a t r e l a t i v e l y h i g h d e n s i t i e s .
But a l l t h e s e d e s c r i p t i o n s a r e v a l i d f o r l o n g polymeric molecules
( S t r a l e y , 1973) a n d t h e p r e d i c t e d S a t t h e n e m a t i c - i s o t r o p i c t r a n s i -
( - 0 . 8 5 ) l e a d i n g t o a n a b r u p t t r a n s i t i o n from
t i o n is q u i t e high
a
strongly- ordered
to a
nematic
completely
disordered
isotropic
phase.
For r e l a t i v e l y s h o r t rods (of length t o bseadth r a t i o - 3 - 5 )
and high d e n s i t i e s t h e scaled p a r t i c l e theory provides a convenient
method
gas.
for
obtaining the
Cotter
(1977)
has
excess f r e e energy
obtained
the
most
relative t o an ideal
complete
form
of
this
theory as applied t o nematic l i q u i d c r y s t a l s .
Another
and
Saupe
type
(1959)
of
mean
according
field
to
theory
which
the
was d e v e l o p e d
anisotropic
by
Maier
dispersion
i n t e r a c t i o n s between t h e m o l e c u l e s a r e r e s p o n s i b l e f o r t h e s t a b i l i t y
of
t h e p h a s e . Each m o l e c u l e was assumed t o be i n a n a v e r a g e o r i e n -
t i n g f i e l d due t o i t s e n v i r o n m e n t ,
its neighbours,
i.e.,
but otherwise uncorrelated with
t h e s h o r t r a n g e o r d e r was i g n o r e d .
Assuming
a c y l i n d r i c a l d i s t r i b u t i o n a b o u t t h e p r e f e r r e d a x i s M a i e r and Saupe
expressed
the
orientational
potential
energy
of
the
molecule
as
w h e r e V i s t h e m o l a r volume a n d A i s a c o n s t a n t d e p e n d i n g o n t h e
molecular species.
Cotter
( 1 9 7 7 ) h a s shown t h a t thermodynamic con-
€ ( c o s 8 ) t o be p r o p o r t i o n a l t o V - I ,
s i s t e n c y demands
v - ~a s
to
r a t h e r than
i n t h e o r i g i n a l Maier- Saupe model.
The n o r m a l i z e d o r i e n t a t i o n a l d i s t r i b u t i o n f u n c t i o n
f ( c o s 8)
z
-1Z
exp [- ~ ( c o 8
s )/kT!
(1.3)
where k i s t h e Boltzrnann c o n s t a n t a n d T t h e a b s o l u t e t e m p e r a t u r e ,
Z t h e p a r t i t i o n f u n c t i o n f o r a m o l e c u l e which i s g i v e n by
Z =
1
exp
0
I-d c o s 8 )IkT]
d(cos
8 ).
(1.4)
The i n t e r n a l e n e r g y i s g i v e n by
w h e r e N i s t h e Avagadro number a n d B
The e n t r o p y i n
=
A/VkT.
relation t o that of
the
isotropic
exp
(? BS c o s 8 )
phase is given
by
S
= -Nk < l n f > =
and
-
1
NkCT BS(2S+1)
t h e Helmholtz f r e e
I
-
In
3
2
d(cos 8 ) 1 (1.6)
energy i n r e l a t i o n t o t h e i s o t r o p i c phase
is
The c o n d i t i o n f o r e q u i l i b r i u m i s
( a ~ / Sa)
V,T
=
0,
which l e a d s
s i s t e n c y demands
to
v-'
E
( c o s 8 ) t o be p r o p o r t i o n a l t o V - I ,
r a t h e r than
a s i n t h e o r i g i n a l Maier-Saupe model.
The normalized o r i e n t a t i o n a l d i s t r i b u t i o n f u n c t i o n
f(cos 8 )
= -1z exp [ - ~ ( c o sB)/kT]
(1.3)
where k i s t h e Boltzmann c o n s t a n t and T t h e a b s o l u t e temperature,
Z t h e p a r t i t i o n f u n c t i o n f o r a molecule which i s given by
Z =
I
exp [ - ~ ( ~8 0)/*TI
s
~ ( C O S8
(1.4)
).
0
The i n t e r n a l energy i s g i v e n by
where N i s t h e Avagadro number and B
z
The e n t r o p y i n r e l a t i o n t o t h a t of
A/VkT.
t h e i s o t r o p i c phase i s given
by
and t h e Helmholtz f r e e energy i n r e l a t i o n t o t h e i s o t r o p i c phase
is
The c o n d i t i o n f o r e q u i l i b r i u m , i s ( 8 F/a S ) V , T
=
0,
which l e a d s
t o t h e following self consistency equation f o r S
This
theory
to a
leads
first
order
nematic- isotropic transition
w i t h a n S v a l u e o f 0 . 4 2 a t t h e t r a n s i t i o n p o i n t TNI. E x p e r i m e n t a l l y
S a t TNI v a r i e s between 0 . 3 t o 0.5 f o r d i f f e r e n t compounds.
Higher
o r d e r terms c a n b e i n t r o d u c e d i n t h e p o t e n t i a l t o a c c o u n t f o r t h e s e
differences
(Chandrasekhar
al.
1972).
But
that
they
t h e main
a
predict
1973).
Madhusudana
drawback of
heat
by a f a c t o r o f 2 o r 3 .
account
and
of
1971,
Humphries
t h e mean f i e l d
transition
which
is
et
t h e o r i e s is
usually
higher
The t h e o r y c a n be improved by t a k i n g i n t o
n e a r n e i g h b o u r c o r r e l a t i o n s (Madhusudana and C h a n d r a s e k h a r ,
However
the
shape anisotropy
of
the
nematogenic
compounds
h a s n o t been t a k e n i n t o a c c o u n t i n t h e s e t h e o r i e s .
A more r e a l i s t i c t h e o r y i n c o r p o r a t e s t h e a t t r a c t i v e p o t e n t i a l
between
hybrid
t h e molecules
models
have
as
been
well as
proposed.
the
hard
Among
rod
features.
these Cotter
Several
(1977)
has
u s e d t h e s c a l e d p a r t i c l e t h e o r y t o d e v e l o p a h y b r i d model a n d h a s
o b t a i n e d t h e c o e f f i c i e n t o f t h e p o t e n t i a l w i t h a t e m p e r a t u r e dependent
attractive
hard
particle
improved
part
and
contribution.
further
by
using
an
athermal
These
the
part
corresponding
theoretical
Andrew's
(1975)
results
model
to
have
the
been
(Savithramma
a n d Madhusudana 1 9 8 0 , Madhusudana 1981 ) .
However
in
a l l t h e s e m o d e l s t h e m o l e c u l e s have been
taken
t o be c y l i n d r i c a l l y symmetric.
In
r e a l i t y most molecules d e v i a t e
from t h i s shape s o t h a t t h e t r a n s v e r s e c r o s s s e c t i o n d e v i a t e s from
a
et al.
of
Alben e t a l .
c i r c u l a r shape.
(19721, S t r a l e y
(19731, Luckhurst
( 1 9 7 5 ) , G e l b a r t and Barboy (1979) have shown t h e importance
taking
this
deviation
from
cylindrical
symmetry
i n t o account
i n the theoretical calculations.
Alben e t a l .
adequate
(1972) have pointed o u t t h a t one o r d e r parameter
is
not
t o describe
t h e u n i a x i a l nematic phase made up
of
p a r t i c l e s w i t h o u t r o t a t i o n a l symmetry and t h a t two o r d e r para-
meters a r e r e a u i r e d .
I n g e n e r a l f o r a molecule of a r b i t r a r y shape t h e o r i e n t a t i o n a l
o r d e r i n t h e nematic phase can be c h a r a c t e r i s e d by t h e t e n s o r
,B
where
and i , j =
tensor
= x , y , z r e f e r t o t h e l a b o r a t o r y f i x e d c o o r d i n a t e system
C,n,<
ZB.
is
1J
r e f e r t o t h e frame l i n k e d t o t h e molecule.
The
r e a l , symmetric and h a s zero t r a c e . If t h e molecular
c o o r d i n a t e system i s chosen such t h a t S a.B. i s d i a g o n a l two indepen1J
d e n t o r d e r parameters which a r e denoted by S and D r e s u l t .
This w i l l
we
also
present
be
discussed
some
i n more d e t a i l i n Chapter 3 where
experimental
results
on
the
determination
o f t h e o r d e r parameter S, u t i l i s i n g t h e I n f r a r e d dichroism method.
As t h e d e v i a t i o n from c y l i n d r i c a l symmetry of
t h e molecules
i n c r e a s e s and t h e molecules become more and more b i a x i a l , one can
expect
that
t h e medium i t s e l f
becomes b i a x i a l .
Freiser
(1970) has
g e n e r a l i s e d t h e i n t e r a c t i o n employed i n t h e Maier-Saupe (MS) t h e o r y
of
the
nematic
stat'e
i n a manner c o n s i s t e n t
with
t h e assymmetry
o f t h e molecules. Such a n i n t e r a c t i o n l e a d s t o a f i r s t o r d e r t r a n s i t i o n from t h e i s o t r o p i c
second o r d e r
transition
Another
phase
to a
possibility
u n i a x i a l phase
followed
b i a x i a l phase a t lower
of
theoretically investigated.
t u r e s of
to a
obtaining
the
by a
temperatures.
bi'axiality
has
been
These t h e o r i e s a r e concerned w i t h mix-
nematogens composed of
r o d - l i k e and p l a t e - l i k e molecules.
When two nematogens made up o f r o d - l i k e molecules a r e mixed t o g e t h e r
u s u a l l y a c o n t i n u o u s m i s c i b i l i t y i n t h e nematic phase i s o b t a i n e d .
The Schroder- Van- laar (Schroder 1893,
mits t h e c a l c u l a t i o n o f
of
the crystal
the
Van-Laar
eutectic
t o nematic t r a n s i t i o n .
1908) e q u a t i o n
temperature
and
per-
composition
A t y p i c a l phase diagram of
m i x t u r e s o f two nematogens i s shown i n F i g . 1.5. T h i s kind of behav i o r i s t r u e f o r most nematic m i x t u r e s .
If one o f
the
nematic- isotropic
depressed.
t h e m i x t u r e i s non-mesomorphic,
t h e components of
Further
the
transition
nematic
a considerable range ( F i g . 1.6)
temperature
and
isotropic
of
the
phases
nematogen
coexist
is
over
( M a r t i r e 1979). We have found t h a t
s p e c i a l t y p e s of d e f e c t s ( t h e high s t r e n g t h d e f e c t s ) occur i n c e r t a i n mixtures of
t h i s type.
d i s c u s s e d i n Chapter
6.
The s t u d i e s on t h e s e d e f e c t s w i l l be
,
increases and the molecules become more and more biaxial, one can
expect that the medium itself becomes biaxial. Freiser (1970) has
generalised the interaction employed in the Maier-Saupe (MS) theory
of the nematic stat'e
in a manner consistent with the assymmetry
of the molecules. Such an interaction leads to a first order transition from the isotropic phase to a uniaxial phase followed by a
second order transition to a biaxial phase at lower temperatures.
Another
possibility
of
obtaining
the
bi'axiality has
been
theoretically investigated. These theories are concerned with mixtures of nematogens composed of rod-like and plate-like molecules.
When two nematogens made up of rod-like molecules are mixed together
usually a continuous miscibility in the nematic phase is obtained.
The Schroder-Van-laar (Schroder 1893, Van-laar 1908) equation permits the calculation of the eutectic temperature and composition
of the crystal to nematic transition. A typical phase diagram of
mixtures of two nematogens is shown in Fig. 1.5. This kind of behavior is true for most nematic mixtures.
If one of the components of the mixture is non-mesomorphic,
the nematic-isotropic transition temperature of the nematogen is
depressed. Further the nematic and isotropic phases coexist over
a considerable range (Fig.
1.6) (Martire 1979). We have found that
special types of defects (the high strength defects) occur in certain mixtures of this type. The studies on these defects will be
discussed in Chapter 6.
.
-
Nematic
-I
I
20
I
I
40
80
P - AZOXYANI SOLE, MOLE 7,
60
I
100
Figure 1.5
T y p i c a l p h a s e d i a g r a m of m i x t u r e s o f two r o d - l i k e n e m a t o g e n s
(From Hsu and J o h n s o n , 1973)
Figure 1.6
Coexistence o f nematic and isotropic phases
in mixtures of mesomorphic compounds with
non-mesomorphic solutes. (From Martire 1 9 7 9 ) .
O f t e n when
terminal
polar
and
terminal
n o n - p o l a r compounds,
b o t h of which e x h i b i t o n l y a n e m a t i c p h a s e i n t h e p u r e s t a t e , are
mixed
Fig.
together a
1.7)
smectic
(For example,
phase r e s u l t s i n
Dave e t a l .
1 9 6 8 , Heppke and R i c h t e r 1978,
d e n c e t h a t t h e o c c u r r e n c e of
m i x t u r e s c a n be a t t r i b u t e d
an
induced smectic
A
t h e mixed s t a t e
1966, S c h r o e d e r and S c h r o e d e r
1978). There i s e v i -
Engelen e t a l .
smectic p h a s e 1 i n some
t h i s 'induced
t o charge
p h a s e h a s been
(e.g.,
t r a n s f e r complexing.
found
in a
However
s y s t e m i n which
n e i t h e r o f t h e components h a s a n a p p r e c i a b l y s t ' r o n g p o l a r end g r o u p
(Suresh 1983).
One o f o u r s t u d i e s is o n b i n a r y p h a s e d i a g r a m s o f two nematog e n s h a v i n g r o d - l i k e and p l a t e - l i k e m o l e c u l e s . Alben ('1973) t h e o r e t i c a l l y s t u d i e d s u c h m i x t u r e s b a s e d on a l a t t i c e model and p r e d i c t e d
the
occurrence
(1982)
arrived
Deutch
(1984)
of
a b i a x i a l phase ( F i g . 1 . 8 ) .
at
similar
using a
Caflisch e t a l .
conclusions.
of
t h e o r y and
Chen
and
that of
( 1 9 8 4 ) a l s o u s i n g a l a t t i c e model, g a v e e s s e n t i a l l y
The f i r s t e x p e r i m e n t a l e v i d e n c e f o r t h i s
b i a x i a l i t y was o b s e r v e d i n a
a c o n c e n t r a t i o n and
phases,
theories
Van d e r Walls t y p e o f
t h e same t y p e o f r e s u l t s .
in
The
L a t e r Rabin e t a1
l y o t r o p i c system
(Yu a n d Saupe 1980)
temperature range s e p a r a t i n g
two
uniaxial
o n e composed o f d i s c - s h a p e d m i c e l l e s a n d t h e o t h e r of
rod-
s h a p e d micelles.
However, when two t h e r m o t r o p i c nematogens,
rod- like molecules
and
the o t h e r
having
o n e o f them h a v i n g
disc- like molecules,
are
-
7.0.5
MOLE FRACTION
5 CBP
Figure 1.7
Phase diagram of two pure nematogens exhibiting
an induced smectic phase. (From Engelen et al.,
1978).
'10 plates
Figure 1.8
Theoretical phase diagram o f mixtures o f rods
and plates indicating the occurrence of a biaxial
phase. i : isotropic,
n - : negative
T* : effective
n(+) : positive uniaxial,
uniaxial,
temperature.
6 : biaxial
(From
Alben
phase,
1973).
mixed
together,
found
a
instead
coexistence
of
of
a
getting
two
nematic
biaxial
phases.
nematic
This
phase,
result
will
we
be
d i s c u s s e d i n C h a p t e r 2.
1.5
CONTINUUM THEORY OF NEMATIC LIQUID CRYSTALS
S i n c e n e m a t i c l i q u i d c r y s t a l s are c h a r a c t e r i s e d by a n o r i e n t a -
tional
order,
any
deformation
by a r e s t o r i n g t o r q u e .
of
the
director
is
field
opposed
The d e f o r m a t i o n s a r e d e s c r i b e d by t h e c o n t i -
nuum t h e o r y . Oseen ( 1 9 3 3 ) and Z o c h e r l(1933) i n i t i a t e d t h e c o n t i n u u m
model which was reexamined by F r a n k ( 1 9 5 8 ) .
These
the
intermolecular
deformation
by
deformations
a
define
linear
total
of
elastic energy.
forces are additive
dependent
as a
him
cost
part
of
the
combination
eight
elastic
assumed
and have s h o r t
energy
of
Oseen
ten
density
terms
constants.
range.
was
whose
Five o f
that
The
expressed
coefficients
these
terms
and t h r e e o f t h e e l a s t i c c o n s t a n t s do n o t e n t e r t h e Euler- Lagrange
equation
determines the possible equilibrium s t r u c t u r e s of
*
t h e l i q u i d c r y s t a l . They were t h u s o m i t t e d by Oseen ( 1 9 3 3 ) . F r a n k
derived
which
t h e t h e o r y i n a p h e n o m e n o l o g i c a l a p p r o a c h a n d some of
the
t e r m s d r o p p e d by Oseen were r e i n t r o d u c e d by him.
Following t h e formalism o f Frank a n e x p r e s s i o n f o r t h e e l a s t i c
f r e e e n e r g y d e n s i t y c a n be o b t a i n e d a s f o l l o w s .
t h e preferred
o r i e n t a t i o n a t any point
Let % ( r ) represent
r. It i s assumed t o v a r y
s l o w l y w i t h p o s i t i o n and i s d e f i n e d a t o t h e r p o i n t s by c o n t i n u i t y .
At
any
point
r a r i g h t handed C a r t e s i a n c o o r d i n a t e s y s t e m x , y , z
+
w i t h z - a x i s a l o n g n i s i n t r o d u c e d . The x - a x i s i s a r b i t r a r i l y c h o s e n
b u t i s t a k e n t o be p e r p e n d i c u l a r t o z and y- axes. The s i x components
o f c u r v a t u r e ( s e e F i g . 1 . 9 ) a r e g i v e n by:
where S, t a n d b d e n o t e s p l a y ,
tively.
Frank
Considering
the
t w i s t a n d bend d e f o r m a t i o n s r e s p e c -
symmetry
(1958) h a s shown
that
of
the
uniaxial
nematic
t h e f r e e energy d e n s i t y
phase
Fd i s g i v e n
by
where K 1 , ,
K22
and K
s t a n t s associated
the
nematic
crystal
the
free
can
energy
a r e t h e s p l a y , t w i s t and bend e l a s t i c con-
with
(Fig.
be
33
the
1.9).
three
The
determined
density.
bulk
distortions
equilibrium s t r u c t u r e of
by
A
independent
minimising
necessary
t o have a n e n e r g y minimum i s t h a t
% is
the
volume
condition
for
a solution of
the
liquid
integral
a
of
of
structure
t h e Euler-
6 nx,
J
X
SPLAY
TWIST
BEND
Figure 1.9
The t h r e e t y p e s o f d e f o r m a t i o n
i n a nematic l i q u i d
c r y s t a l (From F r a n k 1958).
As we have s e e n Fd i s a f u n c t i o n
Lagrange d i f f e r e n t i a l e q u a t i o n .
of
and i t s g r a d i e n t s .
where x
n
.
z
1
:
x,
z
y l = y ,
1
and
= z
+
n
I
= n
x'
n
2
--
n
Y
andn
3
=
I m p o s i n g a small v a r i a t i o n 6 n ( r ) a t a l l p o i n t s we have
Integrating
the
second
term
by
p a r t s and n e g l e c t i n g
the surface
term, i . e . ,
assuming t h a t t h e a n c h o r i n g a t t h e b o u n d a r i e s i s s t r o n g
T h i s y i e l d s t h e Euler-Lagrange e q u i l i b r i u m e q u a t i o n
+
+
Terms o f t h e form d i v u , where u ( r ) i s a n a r b i t r a r y v e c t o r f i e l d ,
may be
transformed
t o s u r f a c e i n t e g r a l s u s i n g t h e Gauss theorem.
These t e r m s do n o t c o n t r i b u t e t o t h e e q u i l i b r i u m c o n d i t i o n ( 1 . 1 4 ) .
For
example
the
term a s s o c i a t e d w i t h
F r a n k c a n b e w r i t t e n as ( E r i c k s e n 1962)
(KZ2
+ KZ4) i n t r o d u c e d by
and c a n t h u s be o m i t t e d . F i n a l l y i n v e c t o r n o t a t i o n t h e f r e e e n e r g y
d e n s i t y of t h e n e m a t i c i s u s u a l l y w r i t t e n a s
Fd
1
+
2
= -2 [ K l l ( d i v n ) + K22
(z . c u r l n ) 2 +K 33 (-+n x
+
c u r l -+n ) 2 ]
The f r e e e n e r g y d e n s i t y i s a q u a d r a t i c f u n c t i o n o f
i n t h e d i r e c t o r f i e l d s , i . e . , i n t h e g r a d i e n t s on
Nehring
t h e second
and
Saupe
derivatives
(1971)
of
argued
that
(1.16)
the curvature
h.
the
terms l i n e a r
in
8 can make c o n t r i b u t i o n s o f t h e same
o r d e r of magnitude a s t h e q u a d r a t i c terms o f t h e f i r s t d e r i v a t i v e s .
I f t h i s i s t a k e n i n t o a c c o u n t a term of t h e form div(ff d i v
t e d w i t h K13 s h o u l d be i n c l u d e d i n Fd.
of similar order associated with
(K
22
h)
connec-
Frank who c o n s i d e r e d terms
+ K2,,),
ignored t h i s term.
However a s i n t h e c a s e o f t h e l a t t e r , t h e K
term makes no c o n t r i b u 13
t i o n when t h e r e i s s t r o n g a n c h o r i n g a t t h e b o u n d a r i e s . T h i s problem
w i l l be d e a l t w i t h i n d e t a i l i n C h a p t e r 5 a l o n g w i t h a n e x p e r i m e n t a l
method of d e t e r m i n a t i o n o f t h e e l a s t i c c o n s t a n t K
As
mentioned
by a n a p o l a r
earlier
director.
Most
the
of
nematic
13'
medium
is
characterised
t h e n e m a t i c l i q u i d c r y s t a l s made
up o f r o d - l i k e m o l e c u l e s have a e t i v e d i a m a g n e t i c s u s c e p t i b i l i t y .
Therefore i f a magnetic f i e l d i s a p p l i e d perpendicular t o t h e o r i g i -
7
nal
direction
at
Anchored
of
alignment
the
of
boundaries,
a
t h a t is strongly
n e m a t i c sample
distortions
in
the
uniform
pattern
However as t h e e f f e c t o f t h e m a g n e t i c
c a n be o b t a i n e d ( F i g . 1 . 1 0 ) .
f i e l d i s opposed by t h e s t r o n g a n c h o r i n g a t t h e b o u n d a r i e s ,
distortions
certain
take
place
threshold
only
at
Ha.
value
these
a f i e l d strength. g r e a t e r than a
Distortions
of
this
kind
were
first
s t u d i e d by F r e e d e r i c k s z ( 1 9 3 3 ) .
t h e s t a b l e e q u i l i b r i u m s t a t e c a n be found
A t a given f i e l d
by
minimising
the
director
the
free
profile.
Different
energy
The
geometries
with
respect
expression
for
a s shown i n
measuring t h e t h r e e e l a s t i c c o n s t a n t s K
the
Fig.
11'
t o the
variations i n
critical
field
is
1 . 1 0 c a n be u s e d f o r
Kz2 a n d K
33'
H
c
is u s u a l l y
measured by e m p l o y i n g a n o p t i c a l t e c h n i q u e f o r d e t e c t i n g t h e d i s t o r t i o n ( s e e f o r e . g . , d e Gennes 1975, C h a n d r a s e k h a r 1 9 7 7 ) .
Typically
the
values
of
K..
is
the
three
11
of
the
o r d e r of
elastic
-6
10
constants
dynes.
for
For
example
para- azoxyanisole
a t 120°C a r e :
K11
K22
K33
The
Kii
s
-6
0 . 7 ~ 1 0 dynes
=
0 . 4 3 x 10
=
1 . 7 x 10
values
-6
-6
dynes
dynes.
decrease
rather
s t r o n g l y when
T increases.
(They
Figure 1.10
The three geometries used in Freedericksz transition experiments
(From Chandrasekhar 1977).
vary roughly like the square of the order parameter.) Since the
three elastic constants are of the same order of magnitude, it
is often very convenient to treat problems of curvature distortion
if the three elastic constants K1,, KZ2 and K
33
are taken to be
equal (one constant approximation).
In Appendix I we present revised absolute values of the elastic constants K1
and K
of several cyano biphenyls which
K22
33
were originally measured using experimental techniques mentioned
above.
1.6
DEFECTS
In liquid crystals various defects are easily visible under
the microscope even
under low magnification. These defects are
stabilised by boundary conditions. They also occur when there is
a disturbance of the orientation caused by electric or magnetic
fields, temperature gradients and mechanical stresses.
The term
'disclination' was suggested by Frank to describe
-f
defects caused by discontinuities in the director field n(r). There
are two types of disclinations, line and point disclinations.
Linear defects occur frequently in nematic liquid crystals.
They can be visualised by using the Volterra process (Volterra
1907),
Volterra Process
A perfect medium is cut along an arbitrary surface S and
the two lips S f and S" of the cut surface are displaced relative
to each other by a symmetry permitted rotation around a certain
axis. The void which now exists is filled with perfect material,
or if there is an overlap of the two parts the excess of the material is removed and the medium is allowed to relax such that the
director continuity is still maintained. This pr'oduces a line discontinuity along a line L (see for example fig. 1.11).
The
most
frequently occurring
types of disclinations are
those in which the rotation axis is parallel to the line L. These
are called
wedge disclinations. The Schlieren
texture exhibited
by nematics is due to these wedge disclinations. If the rotation
is perpendicular to the singular line, we get a twist disclination.
In this case the line of singularity is in the plane in which the
molecules lie. These are often seen as loops separating regions
of different twist.
Schlieren Textures
When a sufficiently thin (-20
)
nematic sample is taken
between glass surfaces that have not been treated to give any preferred alignment at the boundaries, one oftqn sees a set of singular
points. These correspond to discontinuities in the director orientation and are disclination lines viewed end on. When the sample
is taken between crossed polarisers these points are seen to be
Figure 1.11
Generation of a disclination line by
Volterra process.
connected
by b l a c k
brushes.
This
t e x t u r e is c a l l e d
the Schlieren
t e x t u r e . A t y p i c a l example i s shown i n F i g . 1.12.
The b r u s h e s a r e r e g i o n s where t h e d i r e c t o r i s e i t h e r p a r a l l e l
or perpendicular t o t h e plane of polarisation of the incident l i g h t .
The
p o l a r i s a t i o n i s unchanged
the
light
points
is
are
extinguished
associated
by
by
t h e sample i n t h e s e r e g i o n s and
the
either
with
crossed
four
analyser.
brushes
or
Usually
two
the
brushes.
I f the p o l a r i s e r s a r e r o t a t e d simultaneously t h e brushes a l s o r o t a t e
c o n t i n u o u s l y showing t h a t
continuously
about
The s e n s e of
(positive
the
t h e o r i e n t a t i o n of
points,
r o t a t i o n may
disclinations)
which
t h e d i r e c t o r changes
themselves remain
unchanged.
be
t h e same a s t h a t o f
the polarisers
or
opposite
disclinations).
(negative
The r a t e o f r o t a t i o n i s a b o u t e q u a l t o t h a t o f t h e p o l a r i s e r s when
t h e r e a r e f o u r b r u s h e s a n d twice as f a s t when t h e r e a r e o n l y two
brushes.
The
by a
of
different
number
brushes).
's'
types
called
The sum of
of
the
s of
disclinations
'strength'
can
defined
be
characterised
as 1 / 4
X (number
a l l d i s c l i n a t i o n s i n a sample t e n d s
t o be z e r o . D i s c l i n a t i o n s w i t h o p p o s i t e v a l u e s a r e a l w a y s c o n n e c t e d
togekher.
Often
near
the nematic- isotropic transition
d i s c l i n a t i o n s merge t o g e t h e r .
point,
the
If t h e sum o f t h e i r s v a l u e s e q u a l s
zero they t o t a l l y a n n i h i l a t e each other.
If on t h e o t h e r hand t h e
s v a l u e s a d d up, a new d i s c l i n a t i o n w i t h a n s v a l u e e q u a l t o t h e
sum, r e s u l t s
F i g u r e 1.12.
A typical Schlieren texture ext1ibited.b~
nematic l i q u i d c r y s t a l s .
The s i g n i f i c a n c e
of
these
Lehmann
(1890)
and
of
actual
configuration
the
by Oseen
Friedel
(1933) a n d Frank
t e x t u r e s was f i r s t u n d e r s t o o d
(1922),
but
around
(1958).
the
a
mathematical
disclinations
A more
by
treatment
was
given
thorough d i s c u s s i o n o f
t h e S c h l i e r e n t e x t u r e is due t o Nehring and Saupe ( 1 9 7 2 ) . The c o n t i nuum t h e o r y o f e l a s t i c d e f o r m a t i o n s forms t h e b a s i s of s u c h t h e o r e t i -
cal
treatments.
Assuming a
one
constant
approximation,
solutions
t o t h e d i r e c t o r f i e l d around d i s c l i n a t i o n s c a n e a s i l y be o b t a i n e d .
If
that
+
$ i s t h e a n g l e made by n w i t h t h e x - a x i s , and a i s t h e a n g l e
the
r a d i u s vector connecting
t h e given
point
o f t h e d e f e c t makes w i t h t h e x - a x i s , and assuming t h a t
t o the centre
*n
is confined
t o t h e x y p l a n e t h e s o l u t i o n s t a k e t h e form
where c i s a c o n s t a n t .
I n t h i s approximation
$
becomes m u l t i p l e
v a l u e d a t t h e o r i g i n and hence l e a d s t o a s i n g u l a r i t y a t t h a t p o i n t .
Assuming
energy
that a
per
unit
core
region extends
length
of
an
from t h e o r i g i n
isolated
to r
c'
d i s c l i n a t i o n is g i v e n
the
by
( N e h r i n g a n d Saupe 1972)
where Wc a r i s e s from t h e c o r e r e g i o n and R is t h e s i z e o f t h e sample.
For a g i v e n R t h e e l a s t i c e n e r g y c a r r i e d by a d e f e c t i s t h u s s e e n
t o be p r o p o r t i o n a l
to s
2
which means t h a t a l l d e f e c t s w i t h
>1/2 s h o u l d s p o n t a n e o u s l y b r e a k up t o form d e f e c t s w i t h
However d e f e c t s o f str
e n t l y seen.
1s
1
1s 1 = I n .
C l a d i s and Kleman (1972) and Meyer (1973) r e s o l v e d t h i s problem and
1s (
:
showed
that
the
energy of
the disolination of
strength
1 i s reduced i f t h e d i r e c t o r i s allowed t o r e l a x o u t of t h e
p l a n e p e r p e n d i c u l a r t o t h e d i s c l i n a t i o n l i n e , such t h a t i t i s a l o n g
the
z- axis a t
the
origin.
T h i s s o l u t i o n has
no s i n g u l a r i t y and
t h e ' l i n e energy does n o t depend on R .
Often p a i r s of d i s c l i n a t i o n s w i t h o p p o s i t e s v a l u e s a r e s e e n .
In
t h e one c o n s t a n t approximation
p a t t e r n around
a
pair
of
the solutions for
d e f e c t s with
the director
p l a n a r d i r e c t o r f i e l d and
o f s t r e n g t h *s, can be w r i t t e n a s
where
a l and
a2 a r e t h e a n g l e s a s d e f i n e d e a r l i e r with r e s p e c t
t o t h e two d e f e c t s .
The c o n s t a n t c determines t h e o r i e n t a t i o n of
t h e d i r e c t o r f a r away from t h e d e f e c t s .
U n t i l r e c e n t l y o n l y d e f e c t s of s t r e n g t h * I and i1/2 had been
observed
i n nematic l i q u i d
crystals.
I n Chapter 6 we r e p o r t
the
f i r s t o b s e r v a t i o n s of d e f e c t s of s t r e n g t h i3/2 and i2 i n thermotrop i c nematic l i q u i d c r y s t a l s .
Inversion Walls
S c h l i e r e n t e x t u r e s may a l s o a r i s e when t h e s u r f a c e s t r y t o
impose a uniform o r i e n t a t i o n on t h e sample,
f o r example when t h e
g l a s s p l a t e s have been rubbed. The alignment i s g e n e r a l l y p a r a l l e l
Figure 1.13
Molecular alignment in the vicinity of an
inversion wall of the first kind.
(From Nehring and Saupe 1972).
t o t h e d i r e c t i o n imposed by t h e s u r f a c e s e x c e p t i n t h e r e g i o n of
t h e wall ( F i g . 1 . 1 3 )
( N e h r i n g a n d Saupe 1 9 7 2 ) . I n t h i s r e g i o n t h e
a l i g n m e n t c h a n g e s c o n t i n u o u s l y w i t h a change o f
The
t o t a l change
n/2 a t t h e c e n t r e .
i n c r o s s i n g t h e wall corresponds
the preferred orientation
by n.
Between c r o s s e d
t o a t u r n of
polarisers
two
b l a c k b r u s h e s a p p e a r o n e on e i t h e r s i d e o f t h e c e n t r a l l i n e ( n / 2
l i n e ) . One b r u s h o f e a c h p a i r i s more d i f f u s e t h a n t h e o t h e r . T h e s e
walls a r e c a l l e d i n v e r s i o n w a l l s o f t h e f i r s t k i n d .
Another t y p e
o f i n v e r s i o n w a l l c a l l e d i n v e r s i o n w a l l of t h e second kind o c c u r s
when m o l e c u l e s a r e t i l t e d o u t o f t h e p l a n e o f t h e bounding s u r f a c e .
T h e s e a r e i n v e r s i o n w a l l s w i t h r e s p e c t t o t h e t i l t d i r e c t i o n . Molec u l e s on d i f f e r e n t s i d e s o f t h e wall a r e t i l t e d i n o p p o s i t e d i r e c t i o n s and on g o i n g t h r o u g h t h e wall a c o n t i n u o u s change i n t i l t ,
angle t a k e s place, with t h e molecules aligned p a r a l l e l t o t h e surf a c e i n t h e c e n t r e o f t h e wall ( N e h r i n g a n d Saupe 1 9 7 2 ) .
The i n v e r s i o n w a l l s may form c l o s e d l o o p s o r s t a r t from d i s c l i nations of
strength iIQ,
of strength *I.
which may be c o n n e c t e d t o d i s c l i n a t i o n s
I n C h a p t e r 6 , we r e p o r t o b s e r v a t i o n s on i n v e r s i o n
w a l l s o f t h e f i r s t kind a s s o c i a t e d with d e f e c t s o f s t r e n g t h
i
3/2.
The n e m a t i c l i q u i d c r y s t a l f l o w s b u t t h e r e i s a c l o s e c o u p l i n g
between t h e flow a n d t h e o r i e n t a t i o n o f
t h e molecules.
The f l o w
c a n i n d u c e a change i n o r i e n t a t i o n which i n t u r n a f f e c t s t h e m o t i o n ,
o r a change i n o r i e n t a t i o n c a n l e a d t o f l o w which t e n d s t o c o u n t e r a c t
o r r e i n f o r c e t h e change o f a l i g n m e n t .
Dynamic
problems o f a n i s o t r o p i c l i q u i d s were f i r s t s t u d i e d
by A n z e l i u s ( 1 9 3 1 ) a n d Oseen ( 1 9 3 3 ) . The c o u p l i n g between o r i e n t a t i o n a n d f l o w f o r n e m a t i c and c h o l e s t e r i c l i q u i d c r y s t a l s i n p a r t i c u l a r was a n a l y s e d by E r i c k s e n
( 1 9 6 6 ) and
Leslie
(1968,
1979).
Q u a n t i t a t i v e s t u d i e s on f l o w p r o p e r t i e s have been made by v a r i o u s
g r o u p s ( s e e d e Gennes 1975; C h a n d r a s e k h a r 1 9 7 7 ) .
As
in
conventional
hydrodynamics
the
dynamical
situation
-P
f o r a n e m a t i c l i q u i d c r y s t a l i s s p e c i f i e d by a v e l o c i t y f i e l d v ( r )
g i v i n g t h e flow of matter, and i n t h e Leslie f o r m u l a t i o n ( L e s l i e
-t
1979), t h e d i r e c t o r n ( r ) d e s c r i b i n g
is s p e c i f i e d
i n addition.
t h e l o c a l s t a t e of alignment
W
e first write an expression
f o r the
d i s s i p a t i o n o r t h e e n t r o p y s o u r c e , due t o a l l f r i c t i o n p r o c e s s e s
i n t h e f l u i d . T h e r e a r e two t y p e s o f d i s s i p a t i v e l o s s e s , c o n v e n t i o nal
viscosity effects
and l o s s e s a s s o c i a t e d w i t h a
r o t a t i o n of
t h e d i r e c t o r w i t h r e s p e c t t o t h e background f l u i d . F o l l o w i n g t h e
n o t a t i o n s o f d e Gennes ( 1 9 7 5 ) t h e d i s s i p a t i o n T$ i s g i v e n by
where
0'
i s t h e symmetric p a r t of
t h e stress t e n s o r , A is t h e
symmetric p a r t o f t h e v e l o c i t y g r a d i e n t t e n s o r g i v e n by
-t
w
1
-t
v,
c
u
r
l
2
:-
is t h e a n t i s y m m e t r i c p a r t o f t h e v e l o c i t y g r a d i e n t
+
tensor,r
::
+
+
n x h i s t h e t o r q u e p e r u n i t volume e x e r t e d by t h e i n t e r -
+
n a l d e g r e e s of freedom.
change
+
of
n per
ti
unit
+
i s t h e m a t e r i a l d e r i v a t i v e of n ,
time a s experienced
i.e.,
by a moving molecule,
+
h i s t h e molecular f i e l d .
Eq. 1 . 2 1 c a n be w r i t t e n a s
+
+
where N = ti
-
+
+
U x n
.
+
The v e c t o r N r e p r e s e n t s . t h e r a t e o f change
of t h e d i r e c t o r w i t h r e f e r e n c e t o t h e background f l u i d .
Eqn. (1.22)
c o n t a i n s t h e two t y p e s of d i s s i p a t i v e l o s s e s due t o v i s c o s i t y e f f e c t s
and r o t a t i o n of t h e d i r e c t o r .
In
the
Ericksen- Leslie
model
the
hydrodynamic
stress
is
assymmetric and h a s i n g e n e r a l s i x v i s c o s i t y c o e f f i c i e n t s .
The c o n t r i b u t i o n
a
sum of
p r o d u c t s of
t o t h e e n t r o p y source i s now expressed a s
f l u x e s and t h e conjugate f o r c e s
( d e Groot
and Mazur 1962). Choosing a s f l u x e s t h e components of t h e symmetric
tensor A
logical
+
a6
and t h e components of t h e v e c t o r N
equations
can be w r i t t e n .
expressing
the
forces
!J'
a s e t of phenomeno-
i n terms o f
the fluxes
Usually one is concerned with very slow motions
s o t h a t t h e f l u i d can be t r e a t e d a s incompressible.
In the l i m i t
o f slow motions t h e f l u x e s a r e l i n e a r f u n c t i o n s of t h e thermodynamic
forces.
Cross e f f e c t s between v a r i o u s phenomena e x i s t , s i n c e each
f l u x may i n p r i n c i p l e be a l i n e a r f u n c t i o n of a l l thermodynamic
forces.
S p a t i a l symmetries of
t h e nematic reduces
t h e number of
phenomenological
coefficients.
The
Onsager
reciprocity
theorem
gives rise to additional relationships amongst the phenomenological
coefficients thus reducing the
number of independent viscosity
coefficients. Thus the following equations are obtained for the
complete viscous stress a '
a6
+
and the hydrodynamic molecular field
h (de Gennes 1975)
and
together with the relations
The six coefficients a . are usually called the Leslie coefficients
1
and have the dimensions of viscosity. Equation (1.26) was first
derived by Parodi (1970) using the Onsager principle. This reduces
the number of independent coefficients to five.
The. cholesteric liquid crystal is a chiral system consisting of orientationally ordered optically active anisotropic molecules, with the director twisting about an orthogonal axis. In
parity conserving systems the Curie (or
Von Neumann) principle
requires that the symmetry of physical effects must be contained
i n t h e c a u s e s which g i v e r i s e t o them. But i n c h i r a l systems which
l a c k mirror,symmetry t h e p r i n c i p l e p r e d i c t s novel t y p e s of c o u p l i n g
between f l u x e s and f o r c e s .
Hence i n c h o l e s t e r i c s a
p o s s i b i l i t y now e x i s t s of a coupl-
i n g between thermal and mechanical e f f e c t s .
mechanical e f f e c t
d i s c o v e r y of
drops t a k e n
was observed
by Lehmann
(1900)
thermo-
soon a f t e r
the
He found t h a t i n s m a l l c h o l e s t e r i c
liquid crystals.
between
T h i s t y p e of
two g l a s s s u r f a c e s a thermal g r a d i e n t a l o n g
t h e h e l i c a l a x i s g i v e s r i s e t o a continuous r o t a t i o n of t h e s t r u c t u r e about
that axis.
The
a n g u l a r momentum d e n s i t y of
force has the character
out a
detailed
thermomechanical e f f e c t
t h e d i r e c t o r even though t h e a p p l i e d
of a
hydrodynamic
r e s u l t s i n an
polar vector.
t h e o r y of
Leslie
(1968)
worked
t h e c h o l e s t e r i c s t a t e and
o b t a i n e d s o l u t i o n s corresponding t o t h e Lehmann r o t a t i o n phenomenon.
L a t e r Martin e t a 1 (1972) and Lubensky (1972) developed a hydrodyna-
\
mic t h e o r y of l a y e r e d systems and showed t h a t t h e symmetry of such
systems
allows
for
a
thermomechanihal
effect
which
couples
the
phase of t h e l a y e r s with a temperature g r a d i e n t .
To o u r knowledge t h e Lehmann experiment h a s never been reproduced.
We have s t u d i e d an analogous e f f e c t
-
the electromechanical
c o u p l i n g i n a c h o l e s t e r i c , under t h e a c t i o n of a D C e l e c t r i c f i e l d .
These s t u d i e s a r e r e p o r t e d i n Chapter 4 i n d e t a i l .
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I RAMAN RESEA1CH INSTITUT~]